TITLE: Spin cobordism v.s. KO theory in low or in any dimensions
QUESTION [8 upvotes]: It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$ 
Question 1:
Then do we have
$$\Omega_d^{spin}(BG)_p = ko_d(BG)_p?$$ for $p=2$ and free part, for $d\le 7$? (how about higher $d>7$?)
And
$$
\Omega_d^{spin}(BG)_p = \Omega_d^{SO}(BG)?
$$
for $p \neq 2$ and $p$ is an odd prime?
Namely, the 2-torsion and free part of $Mspin$ and $KO$ is the same. If there is an odd $p$ torsion, we need to consider localization at odd prime by $MSO$ cohomology. Is this correct?
Question 2:
If this is a statement about the spectra, not just about stable homotopy groups, and thus within these spin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?

REPLY [4 votes]: First of all, let me say that the page you are quoting is a little bit 
misleading if not inaccurate on the Anderson-Brown-Peterson splitting.  
$ko\langle 4n(J)\rangle $ should read $\Sigma ^{4n(J)} ko$  and $ko\langle 4n(J)-2\rangle $ should read $\Sigma ^{4n(J)-4} ko\langle 2\rangle $
See, e.g. https://pdfs.semanticscholar.org/c377/7dd83a6ba0d959c5c62d633ed4109cddb660.pdf 
With this correction, at 2, $Mspin \wedge BG$ splits whose bottom piece is $ko\wedge BG$, other pieces are at least 7-connected since the "next bottom" piese is $\Sigma ^8ko \wedge BG$ as is pointed out by
ArunDebray, corresponding to the partition $J=(2)$.
Thus the answer to your questions, at the prime 2 is that 

We have an isomorphism up to $d\leq 7$
The map is always surjective, but the kernel is in general non-trivial for $d\geq 8$.